Fully packed loop models on finite geometries

Abstract

Fully packed loop models describe the statistics of closely packed nested polygons on the square lattice. Many exact results can be obtained for these models, even for finite geometries, using their close relationship to alternating-sign matrices and the solvable six-vertex and O(n=1) lattice models. Some results for the exact partition function of fully packed loop models on various finite geometries are briefly reviewed, as well as the well-known order-disorder bulk phase transition present in these models. A detailed study is presented of the distribution of boundary nests of polygons in fully packed loop models with mirror or rotational symmetry. The probability distribution function of such nests, as well as the average number of nests, are obtained analytically, albeit conjecturally. It is further shown that fully packed loop models undergo another phase transition as a function of the boundary nest fugacity. At criticality, we derive a scaling form for the nest distribution function which displays an unusual non-Gaussian cubic exponential behaviour.

30 pages, contributed chapter

References in corpus (28)

• Alternating sign matrices and domino tilings
• Another proof of the alternating sign matrix conjecture
• Spin chains and combinatorics
• Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions
• The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions
• Six-Vertex Model with Domain Wall Boundary Conditions and One-Matrix Model
• The Importance of being Odd
• Proof of the Refined Alternating Sign Matrix Conjecture
• Combinatorial nature of ground state vector of O(1) loop model
• Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials
• Spin chains and combinatorics: twisted boundary conditions
• Temperley-Lieb Stochastic Processes
• Symmetry Classes of Alternating Sign Matrices
• Emptiness formation probability of the XXZ spin-1/2 Heisenberg chain at Delta=1/2
• Exact expressions for correlations in the ground state of the dense O(1) loop model
• Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase
• Square ice, alternating sign matrices and classical orthogonal polynomials
• A new way to deal with Izergin-Korepin determinant at root of unity
• Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
• The XXZ spin chain at $\Delta =-1/2$: Bethe roots, symmetric functions and determinants
• Stochastic processes and conformal invariance
• Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon
• Plane partitions II: 5 1/2 symmetry classes
• Critical Q=1 Potts Model and Temperley-Lieb Stochastic Processes
• Boundary qKZ equation and generalized Razumov-Stroganov sum rules for open IRF models
• Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations
• The Rotor Model and Combinatorics
• Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix

Cited by in corpus (3)

• On the weighted enumeration of alternating sign matrices and descending plane partitions
• Diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Multiply-refined enumeration of alternating sign matrices